Abstract
Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocity field computed through a self-consistent interaction potential. Blow up of regular solutions is now well established for such system. In Carrillo et al. (Duke Math J (2011)) [18], a theory of existence and uniqueness based on the geometric approach of gradient flows on Wasserstein space has been developed. We propose in this work to establish the link between this approach and duality solutions. This latter concept of solutions allows in particular to define a flow associated to the velocity field. Then an existence and uniqueness theory for duality solutions is developed in the spirit of James and Vauchelet (NoDEA (2013)) [26]. However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case.
Highlights
Aggregation phenomena in a population of particles interacting under a continuous interaction potential are modelled by a nonlocal nonlinear conservation equation
In the framework of granular media [4, 19, 31], a is the identity function, and interaction potentials are in the form
Since we focus on scalar conservation laws, we can assume without loss of generality that the total mass of the system is scaled to 1 and we will work in some space of probability measures, namely the Wasserstein space of order q ≥ 1, which is the space of probability measures with finite order q moment: Pq(RN ) = μ nonnegative Borel measure, μ(RN ) = 1, |x|qμ(dx) < ∞
Summary
Aggregation phenomena in a population of particles interacting under a continuous interaction potential are modelled by a nonlocal nonlinear conservation equation. The key idea is to use the notion of duality solutions, introduced in [13] for linear conservation equations with discontinuous velocities, where measure-valued solutions arise In that case, this allows to give a convenient meaning to the product of the velocity by the density, so that existence and uniqueness can be proved. The theory developed in [18] is, up to our knowledge, the only one allowing to get existence of global in time weak measure solution for (1.1) in dimension higher than 2 Another possibility could be using the notion of Filippov flow [24], together with the stability results in [10], to obtain a convenient notion of solution to (1.1), following [37]. For the completeness of the paper, a technical Lemma is given in Appendix
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